Section 1.4. Lagrange’s Theorem 33 Each element of a coset aH (e.g., a) is called a representative. The cosets just defined are often called left cosets there are also right cosets of H, namely, subsets of the form Ha = {ha : h ∈ H}. In general, left cosets and right cosets may be different, as we shall soon see. If we use the ∗ notation for the binary operation on a group G, then we denote the coset aH by a∗H, where a∗H = {a∗h : h ∈ H}. In particular, if the operation is addition, then this coset is denoted by a + H = {a + h : h ∈ H}. Of course, a = a1 ∈ aH. Cosets are usually not subgroups. For example, if a / ∈ H, then 1 / ∈ aH (otherwise 1 = ah for some h ∈ H, and this gives the contradiction a = h−1 ∈ H). Example 1.45. (i) If [a] is the congruence class of a mod m, then [a] = a + H, where H = m is the cyclic subgroup of Z generated by m. (ii) Consider the plane R2 as an (additive) abelian group and let L be a line through the origin (see Figure 1.8) as in Example 1.33(ii), the line L is a subgroup of R2. If β ∈ R2, then the coset β+L is the line L containing β that is parallel to L, for if rα ∈ L, then the parallelogram law gives β + rα ∈ L . L L' = β + L r α β β + r α Figure 1.8. The coset β + L. (iii) Let A be an m × n matrix with real entries. If the linear system of equations Ax = b is consistent that is, the solution set {x ∈ Rn : Ax = b} is nonempty, then there is a column vector s ∈ Rn with As = b. Define the solution space S of the homogeneous system Ax = 0 to be {x ∈ Rn : Ax = 0} it is an additive subgroup of Rn. The solution set of the original inhomogeneous system is the coset s + S. (iv) Let An be the alternating group, and let τ ∈ Sn be a transposition [so that τ 2 = (1)]. We claim that Sn = An ∪ τAn. Let α ∈ Sn. If α is even, then α ∈ An if α is odd, then α = τ(τα) ∈ τAn, for τα, being the product of two odd permutations, is even. Note that An ∩ τAn = ∅, for no permutation is simultaneously even and odd.

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